At least two Republican senators agree that authorizing military intervention in Syria should require a supermajority, but have yet to commit to a filibuster, the parliamentary move that would require 60 votes.

“I think it is advisable for the administration to request a 60 vote threshold,” Ron Johnson (R-Wis.), a member of the Senate Committee on Foreign Relations who voted against the panel’s resolution, tells TIME through a spokesperson. “The President needs to demonstrate that he gained the support of the public through a strong vote of support in the United States Senate.”

“There is an incentive for us to force a 60 vote threshold,” said an aide for another Republican senator. “That way it doesn’t pass and we signal to the House that there is not support for this and they don’t take it up.”

As of now, the White House and strike supporters still have a long way to go, a task made more difficult with President Obama overseas meeting with foreign leaders. A whip count by CNN, shows that 24 Senators, including seven Republicans have indicated that they support a strike, while 19 Senators, including five Democrats oppose a strike. In the House, where 217 votes are needed for passage, the numbers look even worse. Already, 115 members have committed to vote no, compared to 23 yes votes. Another whip count by ABC News, which takes into account those judged “likely” to support or oppose the action, show that there are already 217 members against the strike, and just 43 supporting it.

House leaders would prefer the Senate to vote first, and they will likely decline to act if the Senate fails. President Obama or a top White House official has lobbied “at least 60 senators,” according to the Washington Post,revealing the pressure under which the Administration is working.

It is not yet clear which member in the Senate will demand a 60 vote threshold for passing the Syria resolution. Sen. Rand Paul (R-Ky.), who voted against the resolution in committee and filibustered John Brennan’s nomination to lead the CIA earlier this year, has walked back his filibuster threat. “A filibuster is especially inappropriate on a matter pertaining to the use of military force,” says Thomas Mann, a congressional expert for the Brookings Institute. “I don’t think the Republican leadership would support a filibuster.”

Despite authorizing the President to only hit “legitimate military targets in Syria” in a “limited and specified manner” without boots on the ground for 90 days, there are still a number of questions that remain in how the President will be limited that could sink the resolution. For example, Sens. Dick Durbin (D-Ill.) and John McCain (R-Ariz.) have expressed that additional language might be needed to determine what happens if Syrian President Bashar Assad decides to use a chemical weapon on the “91st day,” opening the possibility of a drawn-out conflict.

The resolution also doesn’t define what type of lethal aid it will provide to which “vetted” opposition forces besides the Free Syrian Army, which sometimes coordinates with extremists. Sens. Joe Manchin (D-W.Va.) and Heidi Heitkamp (D-N.D.) are considering a completely new proposal that would give Assad 45 days to sign a chemical weapons ban or face “all elements” of U.S. military power,” according to the Washington Post.

The longer the Senate debates the issue, the less likely the resolution’s chance is for passage, which is why Reid is trying to pass the authorization as fast as possible. The Senate had planned on returning from summer recess on Monday, but Reid has decided to convene a pro forma session to file the resolution at noon on Friday. The legislative process will then start on Monday, allowing a cloture vote Wednesday, and debate to begin on Thursday.